Question

If \frac{d}{dx} \left\\{f\left(x\right)\right\\} = g\left(x\right), then $\int\limits^{b}_{a}$ $f (x)g(x)dx$ is equal to

• A. $\frac{1}{2}\left[f^{2}\left(b\right)-f^{2}\left(a\right)\right]$
• B. $\frac{1}{2}\left[g^{2}\left(b\right)-g^{2}\left(a\right)\right]$
• C. $f(b) - f(a)$
• D. $\frac{1}{2}\left[f\left(b^{2}\right)-f\left(a^{2}\right)\right]$

Answer 1

Answer: (A)

$\frac{1}{2}\left[f^{2}\left(b\right)-f^{2}\left(a\right)\right]$

Answer 2

$f(x)=\int g(x) d x$
$\int\limits_{a}^{b} f(x) g(x) d x=(f(x) \cdot f(x))_{a}^{b}-\int\limits_{a}^{b} g(x) \cdot f(x) d x$
$I=f^{2}(b)-f^{2}(a)$
$I=\frac{1}{2}\left(f^{2}(b)-f^{2}(a)\right)$